Intereting Posts

proof of $\sum_{n=1}^\infty n \cdot x^n= \frac{x}{(x-1)^2}$
Proof of the following fact: $f$ is integrable, $U(f,\mathcal{P})-L(f,\mathcal{P})<\varepsilon$ for any $\varepsilon>0$
How to show that $^{\omega}$ is not locally compact in the uniform topology?
Prove that $\mathrm{Res}=\frac{f(z_0)}{g'(z_0)}$
Show that any prime ideal from such a ring is maximal.
Localizations of $\mathbb{Z}/m\mathbb{Z}$
An element of a group $G$ is not conjugate to its inverse if $\lvert G\rvert$ is odd
I feel that (physics) notation for tensor calculus is awful. Are there any alternative notations worth looking into?
$\|(g\widehat{(f|f|^{2})})^{\vee}\|_{L^{2}} \leq C \|f\|_{L^{2}}^{r} \|(g\hat{f})^{\vee}\|_{L^{2}}$ for some $r\geq 1$?
How to construct isogenies between elliptic curves over finite fields for some simple cases?
Consistency strength of weakly inaccessibles without $\mathsf{GCH}$
How to find basis of $\ker T$ and $\mathrm{Im} T$ for the linear map $T$?
Probability of random integer's digits summing to 12
Finding four numbers
Distribution of sum of iid cos random variables

For a periodic function $f(x)=f(x+T)$, its Fourier transform can be written as an infinite sum:

$$

f(x)=\sum_{-\infty}^{\infty}c_n e^{2\pi i x/T}.

$$

This seems to suggest that the information contained in this periodic function is equivalently contained in this set of coefficients. And the number of the coefficients we need is as many as the number of integers.

- Fourier transform of a compactly supported function
- Imposing Condition on a Cauchy Product
- Fourier coefficient of convex function
- Fourier transform of the Heaviside function
- Absolute convergence of Fourier series of a Hölder continuous function
- Dirichlet problem in the disk: behavior of conjugate function, and the effect of discontinuities

For a function that is not periodic

$$

f(x) = \int_{-\infty}^{\infty}f(\xi) e^{-2\pi i \xi x} d \xi,

$$

the number of coefficients we need seems to be as many as the real numbers.

So how should I understand this mapping in turns of mathematical lauguage, for example, what property is common between the integer numbers and a periodic function, and between the real numbers and a non-periodic function?

- Convergence of the Fourier Transform of the Prime $\zeta$ Functions
- How to use the Spectral Theorem to Derive $L^{2}(\mathbb{R})$ Fourier Transform Theory
- Instructive proofs in functional analysis
- How is the Inverse Fourier Transform derived from the Fourier Transform?
- Sobolev space $H^s(\mathbb{R}^n)$ is an algebra with $2s>n$
- $C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$
- Find the solution of the Dirichlet problem in the half-plane y>0.
- How does 2D spatial Fourier (kx-ky) transform result responds to rotation of the original?
- For a trigonometric polynomial $P$, can $\lim \limits_{n \to \infty} P(n^2) = 0$ without $P(n^2) = 0$?
- Decomposing a discrete signal into a sum of rectangle functions

As Qiaochu Yuan pointed out, both correspondences (circle $\to$ integers and line $\to$ line) are special cases of the Pontryagin duality for locally compact abelian group.

But perhaps an informal description can be helpful too. The Fourier transform decomposes a function (thought of as a “signal”) into waves of different frequencies. In general, a signal can contain waves of arbitrary frequency; hence, the Fourier transform assigns a number (“amplitude/phase density”) to every real number. But a signal of period $T$ can contains only waves with period $T$, $T/2$, $T/3$, … This is why periodic functions are mapped to a discrete sequence of numbers, describing the amplitude and phase of the aforementioned waves.

- The $n$-th derivative of the reciprocal of a function and a binomial identity
- $\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-…}}}} $, what is the nearest value of $y^2 – y$?
- A Geometric Proof of $\zeta(2)=\frac{\pi^2}6$? (and other integer inputs for the Zeta)
- orthogonal eigenvectors
- A Problem on the Möbius Function
- Proof of the affine property of normal distribution for a landscape matrix
- Embed $S^{p} \times S^q$ in $S^d$?
- Properties of automorphism group of $G={Z_5}\times Z_{25}$
- Show $\zeta_p \notin \mathbb{Q}(\zeta_p + \zeta_p^{-1})$
- How many cardinals are there?
- Shortest path on unit sphere under $\|\cdot\|_\infty$
- How to prove inverse direction for correlation coefficient?
- Proving divisibility of $a^3 – a$ by $6$
- Prime factor of $A=14^7+14^2+1$
- How to solve $\arg\left(\frac{z}{z-2}\right) = \frac{\pi}{2}$